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Metamath Proof Explorer

Theorem excomw

Description: Weak version of excom and biconditional form of excomimw . Uses only Tarski's FOL axiom schemes. (Contributed by TM, 24-Jan-2026)

Ref Expression
Hypotheses excomw.1 ⊢ ( 𝑥 = 𝑤 → ( 𝜑 ↔ 𝜓 ) )
excomw.2 ⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ 𝜒 ) )
Assertion excomw ( ∃ 𝑥 ∃ 𝑦 𝜑 ↔ ∃ 𝑦 ∃ 𝑥 𝜑 )

Proof

Step Hyp Ref Expression
1 excomw.1 ⊢ ( 𝑥 = 𝑤 → ( 𝜑 ↔ 𝜓 ) )
2 excomw.2 ⊢ ( 𝑦 = 𝑧 → ( 𝜑 ↔ 𝜒 ) )
3 1 excomimw ⊢ ( ∃ 𝑥 ∃ 𝑦 𝜑 → ∃ 𝑦 ∃ 𝑥 𝜑 )
4 2 excomimw ⊢ ( ∃ 𝑦 ∃ 𝑥 𝜑 → ∃ 𝑥 ∃ 𝑦 𝜑 )
5 3 4 impbii ⊢ ( ∃ 𝑥 ∃ 𝑦 𝜑 ↔ ∃ 𝑦 ∃ 𝑥 𝜑 )